Modeling Self-Similar Traffic Over Multiple Time Scales Based on Hierarchical Markovian and L-System Models

Traffic engineering of IP networks requires the characterization and modeling of network traffic on multiple time scales due to the existence of several statistical properties that are invariant across a range of time scales, such as self-similarity, LRD and multifractality. These properties have a significant impact on network performance and, therefore, traffic models must be able to incorporate them in their mathematical structure and parameter inference procedures. In this work, we address the modeling of network traffic using a multi-time-scale framework. We describe and evaluate the performance of two classes of hierarchical traffic models (Markovian and Lindenmayer-Systems based traffic models) that incorporate the notion of time scale using different approaches: indirectly in the model structure through a fitting of the second-order statistics, in the case of the Markovian models, or directly, in the case of the Lindenmayer-Systems based models. Two Markovian models are proposed to describe the traffic multiscale behavior: the fitting procedure of the first model matches the complete distribution of the arrival process at each time scale of interest, while the second proposed model is constructed using a hierarchical procedure that, starting from a MMPP that matches the distribution of packet counts at the coarsest time scale, successively decomposes each MMPP state into new MMPPs that incorporate a more detailed description of the distribution at finner time scales. The traffic process is then represented by a MMPP equivalent to the constructed hierarchical structure. The proposed L-System model starts from an initial symbol and iteratively generates sequences of symbols, belonging to an alphabet, through successive application of production rules. In a traffic modeling context, the symbols are interpreted as packet arrival rates and each iteration is associated to a finer time scale of the traffic. The accuracy of the different proposed models is evaluated by comparing the probability mass function at each time scale and the queuing behavior (as assessed by the loss probability) corresponding to measured and synthetic traces generated from the inferred models. The well-known pOct Bellcore trace is used to evaluate the accuracy of the proposed models and fitting procedures. The results obtained show that these models are very effective in matching the main characteristics of the trace over the different time scales and their performances are similar